Laboratory Activity 2

7/21/2019 Laboratory Activity 2

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Name: Mark Eric Andam Course and Year: BSECE IVEE 179.1 Section: H4! "a#orator$ Sc%edu&e: 1:'' ( 4:''

Acti)it$ *: Mat%ematica& Mode&in+ o, -%$sica&S$stems

Abstract%is &a#orator$ acti)it$ is created ,or students to #e ,ami&iar /it%

t%e use o, scri0ts and so&)e and ana&$e mat%ematica& mode&s o,

0%$sica& s$stems /it% t%e use o, Sci&a#.

Students /ere +i)en sam0&e 0ro#&ems to 0ractice )arious

0ro#&em2so&)in+ ,eatures o, Sci&a#. 3ne o, t%ese is t%e 3E too&.

Students are introduced to t%is too& to %e&0 t%em so&)e 0ro#&ems suc%

as mat%ematica& mode&s o, 0%$sica& s$stems. Students /ere t%en +i)en

a 0ro#&em to so&)e to test t%e kno/&ed+e t%e$ +ained a,ter t%e said

acti)it$.

1 Objectives

o +ras0 t%e im0ortant ro&e o, mat%ematica& mode&s o, 0%$sica&

s$stems in t%e desi+n and ana&$sis o, contro& s$stems.

o &earn %o/ Sci&a# %e&0s in so&)in+ suc% mode&s.

2 List of equipment/software

-ersona& Com0uter Sci&a#

3 Deliverables

Sci&a# scri0ts and t%eir resu&ts ,or a&& t%e assi+nments and e5ercises

0ro0er&$ discussed and e50&ained.

Ana&$tica& conc&usion ,or t%is &a# acti)it$.

4 Modelin !"#sical $#stems

4%1 Mass&$prin $#stem Model

Consider t%e ,o&&o/in+ Mass 2S0rin+ s$stem s%o/n in 6i+ure 1 /%ere ' is t%e

s0rin+ ,orce fv is t%e ,riction coe8cient ()t* is t%e dis0&acement and f)t*is

t%e a00&ied ,orce:


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e,errin+ to t%e ,ree2#od$ dia+ram /e +et

Md

2x(t)

d t2 +fv

dx(t)dt

+Kx(t)=f( t) . )1*

%e second order &inear dierentia& e;uation ecti)es #e%ind mode&in+ t%e mass2dam0er s$stem can #e man$ and

ma$ inc&ude:

?nderstandin+ t%e d$namics o, suc% s$stem.

Stud$in+ t%e eect o, eac% 0arameter on t%e s$stem suc% as mass M

t%e ,riction coe8cient + and t%e e&astic c%aracteristic '()t*.

esi+nin+ a ne/ com0onent suc% as dam0er or s0rin+.

e0roducin+ a 0ro#&em in order to su++est a so&ution.

4%2 ,sin $cilab in $olvin Ordinar# Di-erential .quations

iure 1% Mass&$prin $#stem and its ree&+od#

Diaram


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Sci&a# can %e&0 so&)e &inear or non&inear ordinar$ dierentia& e;uations


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function dvdt = f(t,v)

dvdt = (Fa-B*v)/M

endfunction

//initial conditions ! t=0, v=0

t0 = 0

v0 = 0

t = 0"0#$"5

v = ode(v0, t0, t, f) //read references for ode tool

clf

%lot&d(t, v)

4% 4 $econd Order OD.0 Mass&$prin $#stem .(ample

In rea&it$ t%e s0rin+ ,orce andor ,riction ,orce can %a)e a more com0&icated

e50ression or cou&d #e re0resented #$ a +ra0% or data ta#&e. 6or instance a

non&inear s0rin+ can #e desi+ned suc% t%at t%e e&astic c%aracteristic is

K xr (t)

/%ere r D 1. 6i+ure * is an e5am0&e o, a non&inear s0rin+.

In suc% case e;uation


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Md

2x(t)

d t2 +fv

dx(t)dt

+K x r( t)=f(t) )4*

E;uation


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= $5

r = $

//differential equation definition

function %rie = f(t,)

%rie($) = (&)

%rie(&) = -(B/M)*(&)-(/M)*((($))+r)(Fa/M)

endfunction

t0 = 0

t = 0"0#$"5

0 = 0 //set initial %osition

%rie0 = 0 //set initial velocit

= ode(.0 %rie0, t0, t, f)

clf

%lot&d(t, ($,")) // ($,")1 %lots the %osition vs tie

// (n, ")1 to %lot a solution in atri for

// 2ith diension n


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8 Assessment

8%1 Assinment

1. S%o/ and discuss t%e +ra0%s o#tained ,rom t%e sam0&e simu&ations.

In Sam0&e *.' t%e +ra0%

s%o/s t%at t%e 0&ot is increasin+

e50onentia&&$. It %o&ds true since it

uses non&inear and non2uni,orm

s0rin+.

Sam0&e *.'

In Sam0&e 1.' t%e +ra0%

s%o/s t%e 0&ot o, its Ve&ocit$ /it%

res0ect to time. %e 0&ot is a&most

&inear meanin+ t%e s0eed isincreasin+ 0ro0ortiona& to time. It

a&so %o&ds true since t%e s0rin+ in

used is uni,orm.

Sam0&e 1.'


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8%2 .(ercise 1

e,errin+ to t%e mass2s0rin+ s$stem e5am0&e:

1. -&ot t%e 0osition and t%e s0eed o, t%e s$stem #ot% /it% res0ect to time

in se0arate +ra0%s.

*. C%an+e t%e )a&ue o, rto * and . -&ot t%e 0osition and s0eed #ot% /it%

res0ect to time. iscuss t%e resu&ts.

6i+ *.1

6or 6i+ *.1 and 6i+ *.* in #ot% cases t%ere is an eect /%en

t%e )a&ue o, r is a&tered. In terms o, 0osition t%ere is a s&i+%t

reduction at t%e end o, t%e 0&ot. %e c%an+ed is not t%at su#stantia&

com0ared to S0eed. Increasin+ t%e )a&ue o, r c&ear&$ s%o/s a

si+nicant dec&ine. %e S0eed is +reat&$ reduced to/ards t%e end.


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. @it% r 1 )ar$ t%e )a&ue o, '


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8%3 .(ercise 2

Consider t%e mec%anica& s$stem de0icted in t%e +ure.

%e in0ut is +i)en #$ f(t) and t%e out0ut is

+i)en #$ y(t). etermine t%e dierentia&

e;uation +o)ernin+ t%e s$stem and usin+ Sci&a#

/rite a scri0t and 0&ot t%e s$stem res0onse suc%

t%at ,orcin+ ,unction f(t) = 1. "et m = 10 k = 1 and b

= 0.5.

a. %e 0eak dis0&acement o, t%e #&ock is a#out 1%9%#. At time about t 1 t%e out0ut reac%es 0eak dis0&acement.c. I, k = 10 t%e 0eak )a&ue o, $ is about 1%:e&1.

6i+ .*


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d. @%at %a00ens to t%e s$stem /%en bis c%an+edG

;


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Sci&a# Enter0rises .

Documents

Laboratory Activity 2